well im a matematician, and all i can tell you is that 3 has beautifull properties, but only as a number... i guess we matematicians dont worry about that kind of stuff...
anyway, i'd like to point out that every theology has its magical or sacred or whatever numbers, so it's ok for tolkien's world to have them
and, here is what i found... it's very interesting, here is the website:
http://www.quicksilver899.com/Tolkien/Tolkien_Dictionary.html
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The formulas that the Lord of Mordor used have universal foundations. Throughout mythology certain esoteric principles seem to permeate, not the least of which are the value of certain numbers.
Three is a nearly divine number, incorporating opposing principles in 'balance' - the two balanced by the one; it represents the 'family', the power of creation in a mother, a father and their offspring who ostensibly bears the best of the two parents combined. It is fitting - and ironic - that the divinity of three should accrue to the Elves, who were not only born immortal, but were the cherished 'children' of the Valar, and shared nearly equal powers in many things. Indeed, the Elves might have early been divided into three groups, those who elected to depart for Valinor and went - the Vanyar, those who elected to depart for Valinor but were delayed - loosly the Teleri, and those who elected to stay in Middle-earth - the Avari.
Seven could be summed up as a number of 'redemption'; it can be divided into the creative force of four and the divine principle of three. The Dwarves were 'redeemed' by Ilúvatar after Aulë created them covertly prior to the awakening of the Elves; their creation was outside the will of Ilúvatar, and Aulë, in his shame, was prepared to destroy these hapless creatures. He raised his great hammer to crush them while they cowered in fear and begged for mercy. The Supreme Being had compassion on the Dwarves and stayed Aulë's hand. He willed that the Dwarves sleep until after the time of awakening of the Quendi, the first of the Children of Ilúvatar.
Because Aulë was the Vala of crafting and fashioning, the Dwarves were intended to be craftsmen, thus the involvement of the creative force of four; since they were created directly by a Vala, rather than in the process of the will of Ilúvatar, and survived by divine judgment, they enjoyed a special divine destiny, thus the involvement of three.
Nine is a number of man; some modern scholars dispute this evaluation, but a review of ancient cosmologies, including the early Christian church, will confirm this analysis. Nine holds the promise of ten - perfection - just by proximity. The number nine is subject to both six, the only 'evil' number, and three, the divine principle. The nature of man is forever torn between the lusts of pride and flesh on the one hand, and the noble values of a divine nature on the other.
Thus Sauron, through Professor Tolkien, assigns three rings to the Elves, seven to the Dwarves and nine to men. If the Elves would bring an alliance of their three rings with the nine rings of men, the combined power would equal twelve; twelve is a worldly number, a number of material completion. The combination would provide considerable power to the alliance, but not a match for the supernatural power of one: the One Ring.
One is a number of unity - everything in existence in one principle; but it is also a number of chaos, especially to a worldly perspective. No worldly force can stand in the wake of one. No worldly power can reconcile the chaos of one.
An alliance between the three rings of the Elves and the seven rings of the Dwarves would bring forth the power of ten; ten is a number of divine perfection. It is philosophically questionable whether the worldly races could attain to that number; but Tolkien, the master story spinner, interdicted such a possibility by setting a stubborn enmity between the two races. The Dwarves were a crabby covetous race, yet able to craft objects of great beauty and skill. It is said in one place that the Dwarves could craft great beauty, but they could not conceive of it. They crafted as the Elves designed and directed.
In the First Age the Elven Sindarin King, Elu Thingol, husband of the divine Maia, Melian, commissioned the Dwarves to set one of the Silmaril jewels into a golden necklace already bedecked with gems from the Undying Lands in the West by the Noldor. The Dwarves conceived an evil covetousness for the stunning piece - the Nauglamir - and slew Thingol in his palace of Menegroth - the 'Thousand Caves' - and escaped with the necklace towards their fortress of Nogrod in the mountains of the Ered Luin in the East. However the Elves swiftly regrouped and retook the Nauglamir ere the Dwarves had traveled far. They returned it to the distressed and grieving Melian. Then the Dwarves raised a great force at Nogrod and marched on Doriath, the kingdom of King Elu Thingol.
Melian had already yielded up her natural body and returned to the Eternal Lands to grieve her loss, thus lifting her divine protection over Menegroth and the kingdom of Doriath. The Dwarves descended on Menegroth with great vengeance, and great was the carnage of that battle. Without the protection of Melian or the courage of Thingol, the Elves of Doriath again lost the Nauglamir to the Dwarves amid terrible losses.
At the Ford of Athrad a combined force of Sindarin and Sylvan Elves, with Beren, son-in-law of Thingol and one of the most valiant of Men of this age, counterattacked and slew the Dwarves and reclaimed the Nauglamir. The few Dwarves that escaped were set upon by Ents as they climbed desperately upwards towards their mountain fortress.
From that time forward enmity festered between Elves and Dwarves, enmity that survived all the way into the last part of the Third Age and the War of the Ring.
Thus Sauron felt confident that he had nothing to fear from any alliance of Dwarves and Elves. When the Dwarf, Gimli, joined the Fellowship of the Ring, he represented the lone exception to this evil calculation by the Dark Lord.
Yet it troubled him, no doubt; most especially while he lacked possession of his necessary instrument: the One Ring.
An alliance of the seven of Dwarves and the nine of Men would equal sixteen, which yields seven , still subject to the power of the Lord of Mordor and the One.
But without his Ring, Sauron could only worry...
Luv ya too though. 
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